Eigenstate Phase Transitions

Bo Zhao

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Adviser: Professor David A. Huse

September 2015

c© Copyright by Bo Zhao, 2015.

All rights reserved.

Abstract

Phase transitions are one of the most exciting physical phenomena ever discovered.

The understanding of phase transitions has long been of interest. Recently eigenstate

phase transitions have been discovered and studied; they are drastically different

from traditional thermal phase transitions. In eigenstate phase transitions, a sharp

change is exhibited in properties of the many-body eigenstates of the Hamiltonian of

a quantum system, but not the thermal equilibrium properties of the same system.

In this thesis, we study two different types of eigenstate phase transitions. The first

is the eigenstate phase transition within the ferromagnetic phase of an infinite-range

spin model. By studying the interplay of the eigenstate thermalization hypothesis

and Ising symmetry breaking, we find two eigenstate phase transitions within the

ferromagnetic phase: In the lowest-temperature phase the magnetization can macro-

scopically oscillate by quantum tunneling between up and down. The relaxation of

the magnetization is always overdamped in the remainder of the ferromagnetic phase,

which is further divided into phases where the system thermally activates itself over

the barrier between the up and down states, and where it quantum tunnels. The

second is the many-body localization phase transition. The eigenstates on one side

of the transition obey the eigenstate thermalization hypothesis; the eigenstates on

the other side are many-body localized, and thus thermal equilibrium need not be

achieved for an initial state even after evolving for an arbitrary long time. We study

this many-body localization phase transition in the strong disorder renormalization

group framework. After setting up a set of coarse-graining rules for a general one

dimensional chain, we get a simple “toy model” and obtain an almost purely ana-

lytical solution to the infinite-randomness critical fixed point renormalization group

equation. We also get an estimate of the correlation length critical exponent ν ≈ 2.5.

iii

Acknowledgements

No one can achieve a PhD without others’ help. I am very thankful to have this

opportunity of pursuing the PhD in physics department in Princeton. First of all,

the honor should be given to my advisor, Professor David Huse. It is he who led me

to the palace of physics. Before entering graduate school in Princeton, I was just a

student who had some ability in learning in courses and doing homework. One of the

most important lessons David has shown me is how to tackle a research topic where

there might be no preexisting theory. By doing research with David during these PhD

years , I have experienced both exciting and painful moments. But David has taught

me, through his own attitude, that when doing research, one should keep cautious,

positive and energetic. What typically happens during research is that one keeps

trying and failing until stages where a new method, model or direction are found. I

am very grateful to learn from David, in terms of both his strong physical intuition

and his encouraging personality.

Secondly, I would like to thank my colleagues who have helped me in my academic

life in my PhD years. In David’s group, I have had many meaningful discussions about

physics with Hyungwon Kim and Liangsheng Zhang. Both of them have helped me a

lot in digesting the background physics in our research projects. I would also like to

thank my colleagues, Aris Alexandradinata and Vedika Khemani who have also given

me lots of advice about many academic aspects.

Thirdly, PhD’s life is not all about research. I enjoyed these years in Jadwin with

all of my friends, including, but not limited to, Chaney Lin, Akshay Kumar, Bin Xu,

Yu Shen and many more graduate students who often appeared on the 4th floor in

Jadwin. Without them, my life in Jadwin would be much more lonely and much less

interesting. There are countless many friends who I enjoyed staying with, so I also

would like to thank all of them.

iv

Last, but definitely not the least, I must give special thanks to my family. The

most selfless love and support were consistently given by my parents. Without them,

this thesis would have never been completed.

v

To my parents.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction 1

1.1 Quantum Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Eigenstate Thermalization Hypothesis . . . . . . . . . . . . . . . . . 5

1.3 Many-body Localization . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Eigenstate Phase Transition . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Three ‘Species’ of Schrödinger Cat States in an Infinite-range Spin

Model 12

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Infinite Range Transverse Field Ising Model . . . . . . . . . . . . . . 16

2.2.1 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Energy Level Degeneracy . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Discrete WKB Method . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Eigenstate Thermalization . . . . . . . . . . . . . . . . . . . . . . . . 24

vii

2.5 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 Thermal Activation and Quantum Tunneling . . . . . . . . . . 26

2.5.2 Paired States and Unpaired States . . . . . . . . . . . . . . . 31

2.6 Analytical Calculation of the Thermal Activation and Quantum Tun-

neling Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7 Asymptotic Behaviors of Both Transitions Near Both QCP and

u = 0 ,Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.1 Near QCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.2 Near u = 0 ,Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . 43

2.8 Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Strong Disorder Renormalization Group Approach to Many-body

Localization Transition 54

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 RG Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.1 RG Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 RG Flow of Probability Distributions . . . . . . . . . . . . . . 67

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.1 Fixed Point Distribution . . . . . . . . . . . . . . . . . . . . . 68

3.3.2 Critical Exponent . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.3 Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 95

A Mathematical Proof of the Degeneracy Formula 97

B Estimation of the Magnitude of Disorder 103

B.1 Lower Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.2 Upper Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 107

viii

C Divergence of First Order Derivative of γ(s̄) at s̄ = s̄min 109

D Discussion on Convexity of Function γ(s̄) 112

Bibliography 116

ix

List of Tables

1.1 A list of some properties of the many-body localized phase contrasted

with properties of the thermal phases. Table from [19]. . . . . . . . . 9

x

List of Figures

2.1 The phase diagram of our model. u is the energy per spin and Γ is

the transverse field. The ground state (zero temperature) is indicated

by the black (solid) line, with the quantum critical point (QCP) indi-

cated. The green (dot-dashed) line is the thermodynamic phase tran-

sition (PT) between the paramagnetic and ferromagnetic (F) phases.

There are two dynamical (eigenstate) phase transitions within the fer-

romagnetic phases, indicated by the blue (dashed) lines. See the text

for discussions of the sharp distinctions between these three phases F1,

F2 and F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Two “potential-energy curves” in our model. . . . . . . . . . . . . . . 20

2.3 Constant energy (E) line described by equation (2.25) by setting s̄y = 0

or just by equation (2.22). It determines the WKB turning point xt

when the total spin density s̄ is fixed. . . . . . . . . . . . . . . . . . . 27

2.4 A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄). . . . . . . 29

2.5 A sketch of the difference between thermal activation and quantum

tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xi

2.6 The level-spacing statistics using 100 realizations of H at N = 15 in

phase F1 within the even sector. δ is the ratio between the smaller

level spacing δ< to the larger level spacing δ> for three consecutive

eigenenergies in the even sector. f is the relative frequency for each

bin in this histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.7 Averages of log (D) in phases F1 and F3, respectively, where D is the

‘eigenstate distance’ defined in the text.. The energy density range we

used in F1 is from the first excited state in each sector up to uc − 0.02

where uc is the energy density at the phase boundary between F1 and

F2, whereas in F3 we used the phase’s full energy density range. N

is the total number of spins varying from 8 to 15. The exponential

decrease of D with increasing N indicates thermalization. The error

bars come from averaging over 100 realizations. . . . . . . . . . . . . 48

2.8 The mean ᾱn and the standard deviation ∆αn of the quantity αn de-

fined in Eq. (2.81). The number of realizations is 1600 for N = 11

(blue dash-dotted lines) and 100 for N = 15 (red dashed lines). The

green (solid) line gives the theoretical quantity α(u,Γ) defined in Eq.

(2.32) for the system size N → ∞. . . . . . . . . . . . . . . . . . . . 49

3.1 A sketch of typical RG moves. For example, (a) is to fuse two adjacent

thermal (T) blocks into one thermal block, called “TT move”; all others

are similar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Q∗(η) up to η = 20 using composite Trapezoidal rule. (a) shows the

curve and (b) demonstrates it in a semi-log plot with a linear regression

fit. (c) plots the cumulative area under the curve. . . . . . . . . . . . 80

3.3 The eigenfunction f−(η) corresponding to eigenvalue 0.3995: (a) Linear

scale (b) Absolute value on log scale. . . . . . . . . . . . . . . . . . . 90

xii

3.4 The eigenfunction f−(η) using either the numerical integration directly

or the diagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5 The cumulative distribution function (Cdf) for both the T-blocks and

I-blocks as the total number of blocks N decreases from 107 to 103 and

the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.6 The difference of cumulative distribution functions (Cdf) between the

T-blocks and I-blocks as the total number of blocks N decreases from

107 to 103 and the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . 94

D.1 Comparison between numerical and analytical result of the thermal ac-

tivation and quantum tunneling transition line; the dots are numerical

points and the line underneath the dots is the analytical result given

by equation (2.47). They fit perfectly well. . . . . . . . . . . . . . . . 113

D.2 Numerical results of ∂γ∂s̄

as a function of s̄ by sowing different pairs of

Γ and u randomly within the entire ferromagnetic phase region. . . . 114

xiii

Chapter 1

Introduction

Our understanding of the laws of nature has taken giant leaps in the past several

centuries, beginning from the time of Galileo. For mechanics, Newton first postulated

the classical equations of motion that every macroscopic object should obey, these

equations, we now call “Newton’s Laws”. These laws solve nearly every mechani-

cal phenomenon we observe in daily life! However, when we would like to further

understand the microscopic structures of thermal behaviors, we must consider many

interacting degrees of freedom, too many to explicitly solve for the dynamics, say

from Newton’s Laws or from Schrödinger equation for a quantum system. This gave

birth to the area of statistical mechanics.

Classical statistical mechanics is very successful at explaining equilibrium ther-

mal behavior in daily life, for example the concept of temperature. Thus, at least

phenomenologically speaking, we know the solution based on the essence of classical

equilibrium statistical mechanics: the ensemble theory. But the foundation of the

ensemble theory is to assume a probabilistic nature of so-called microstates. If we

want to get the equilibrium probability distributions in ensemble theory purely from

Newton’s Laws, either we assume that the initial state is itself an ensemble, or we get

the ensemble by including all long time states of the system (ergodicity). Fortunately,

1

people have successfully discovered a more fundamental theory, quantum mechanics,

which is itself a probabilistic theory. Quantum statistical mechanics is the product

of combining quantum mechanics and statistical mechanics. In quantum statistical

mechanics, ensemble theory naturally arises from the probabilistic nature of quantum

mechanics, so the equilibrium ensemble can emerge at any specific long time even if

the initial state is a pure state.

In quantum statistical mechanics, one of the most fundamental questions is

whether a closed quantum system equilibrates. This question is by no means as easy

as it appears. One of the difficulties is the apparent time reversal symmetry breaking

which arises from low entropy initial states. However, the quantum mechanical

formalism obeys the time reversal symmetry, through the unitary time evolution of

a quantum state. Thus, at least, we need to formulate a more rigorous definition of

equilibration called quantum thermalization.

In the remaining sections of this chapter, I will briefly review the concepts of

quantum statistical mechanics, including, but not limited to, quantum thermalization,

eigenstate thermalization hypothesis and many-body localization.

1.1 Quantum Thermalization

Consider a closed quantum system with a time independent Hamiltonian H. In

quantum mechanics, the system is described by a ket vector |ψ〉 in the Hilbert space.

But in quantum statistical mechanics, |ψ〉 is not enough. It only covers the pure

states, a subset of the quantum states described by the probability operator (also

known as density matrix) ρ. As a probability operator, ρ needs to satisfy the following

constraints:

ρ† = ρ , tr ρ = 1 , (1.1)

2

and also, the spectrum of ρ is in the interval [0, 1].

After describing the quantum state of the system, what we need next is the law

of time evolution. If we choose the Schrödinger picture, the time evolution of ρ(t) as

a function of time t is given by the unitary operator e−iHt/�:

ρ(t) = e−iHt/�ρ(0)eiHt/� , (1.2)

or the differential equation form:

i�d

dtρ(t) = [H, ρ(t)] . (1.3)

By establishing both the quantum state and the rule of time evolution, we are

able to formulate a definition for quantum thermalization. Roughly speaking, as

mentioned earlier, thermalization means that the system goes to equilibrium in the

limit of a long time evolution. But if we consider the full probability operator of an

entire closed quantum system, the time evolution is unitary; it will never reach equi-

librium, because such a state is, by definition, fully determined by initial conditions.

In other words, the unitary time evolution “remembers” all information about the

initial state. Instead, we consider thermalization in terms of subsystems. And the

idea is to use the rest of the full system to serve as a heat bath to thermalize the

selected subsystem.

When we are talking about thermalization, we need to send our system to the

thermodynamic limit where the total degrees of freedom N goes to infinity. In gen-

eral, the full system can have several extensive conserved quantities, for example,

total energy, particle number, total spin, etc. But to keep the discussion simple, we

assume the system only has one extensive conserved quantity, the total energy. So if

the system goes to thermal equilibrium, the equilibrium state has only one thermody-

namic parameter, the temperature T . In order to take the thermodynamic limit, we

3

need a sequence of both the initial states ρN(t = 0) and the Hamiltonian HN labeled

by N . In addition, we need to restrict to initial states ρN(0) where the uncertainty in

total energy is subextensive, so T has zero uncertainty for N → ∞. Now we consider

a subsystem S. The degrees of freedom in the full system F can be decomposed

into the degrees of freedom in the subsystem S and the remaining degrees of freedom

—called the “bath”— B:

F = S ⊕ B , (1.4)

for the sets of degrees of freedom. (It is a tensor product in Hilbert space.) Then

the thermodynamic limit corresponds to taking both F and B to infinity but keeping

S finite. We consider the probability operator ρS(t) on S (known as the “reduced”

density matrix)

ρS(t) = trB ρN(t) . (1.5)

Thermalization on the subsystem S means that by sending the system to the ther-

modynamic limit, the probability operator of the subsystem S goes to its thermal

equilibrium value indicated by the temperature T :

limN→∞t→∞

ρS(t) = ρeqS (T ) , (1.6)

where

ρeqS (T ) = lim

N→∞trB ρ

eqN (T ) (1.7)

4

and

ρeqN (T ) =

1

Ze−βH . (1.8)

The last expression is the standard quantum statistical mechanical formula describing

the equilibrium state in the canonical ensemble where Z is the partition function

Z = trF e−βH and β is the temperature parameter β = 1/kBT .

In other words, quantum thermalization of a subsystem S means that the subsys-

tem behaves as if the full system is exactly at the thermal equilibrium state. And

the rest of the full system serves as a heat bath. And if for all subsystems S, at long

time t the above thermalization condition is true for the same temperature T , and

for all initial states corresponding to that T , we say the system thermalizes for this

temperature. [10, 26, 27, 22, 19]

1.2 Eigenstate Thermalization Hypothesis

By defining quantum thermalization, we can further study the destiny of a given

closed quantum system H. From everyday thermal phenomena, one would expect

that if a system can equilibrate, then the system must thermalize from any initial

conditions. Then, it follows that all the exact many-body eigenstates of the full

system must be thermal because the probability operators, ρeigen = |ψ〉〈ψ|, do not

change with time, where |ψ〉 satisfies H|ψ〉 = E|ψ〉. This motivates the following

hypothesis called the Eigenstate Thermalization Hypothesis (ETH):[10, 26, 27, 22]

Every single eigenstate thermalizes.

By saying eigenstate, we are making no approximations. We mean the exact many-

body eigenstate of the full system. By saying thermalize, we mean all subsystems

thermalize for the temperature determined by the eigenenergy of the eigenstate.

5

ETH is a hypothesis. It is not true for systems that are many-body localized

(MBL) as I will discuss in the next section. Also, it is not true for integrable systems

which contain infinite number of local conservation laws. By saying local, we do

not necessarily mean local in real space, it can be local in for example momentum

space. But it should not be global because it is trivial that for any given full system

eigenstate |n〉, the projection operator |n〉〈n| is a conserved operator. And so is any

weighted sum over them. Thus there are infinite number of “global” conservation

laws for any given system. But these projection operators are global and presently

inaccessible to measurement.

ETH is a hypothesis in another sense, that even for those systems where ETH

appears to be true, for example everyday thermal phenomena, it is extremely hard to

prove, even numerically, because one needs to test the exact many-body eigenstates

which requires the exact diagonalization of the full system. In practice, this approach

is limited to N only up to about 20, due to the exponential growth of the dimension

of H. But that has not deferred people from finding numerical evidence for ETH.

Indeed, there has been plenty of research tackling the numerical test of ETH and

there is strong support that ETH is true for a large number of systems, for example

Rigol, Dunjko and Olshanii [22]; Pal and Huse [20]; Kim and Huse [16]; Beugeling,

Moessner, and Haque [6] and many more.

Thus it is worthwhile to mention some consequences based on ETH. One of them

is for the diagonal ensemble. We write the probability operator in the basis of the

exact many-body eigenstates of the full Hamiltonian:

ρ =∑

m,n

|m〉ρmn〈n| , (1.9)

then based on the time evolution of the probability operator, we have that the diagonal

matrix element ρnn stays constant and the off-diagonal matrix element ρmn, m = n,

6

evolves by multiplying a “simple” phase factor:

ρmn(t) = e−i(Em−En)t/�ρmn(0) , m = n , (1.10)

where Em and En are the eigenenergies of the corresponding eigenstates |m〉 and |n〉:

H|n〉 = En|n〉 . (1.11)

When t → ∞, the phase factors in the off-diagonal terms become essentially random

and when they contribute to the probability operator of any local subsystem, they

effectively go to their mean value 0. This procedure is called dephasing which causes

the full system to thermalize when ETH is true. After dephasing, the probability

operator becomes diagonal:

ρD =∑

n

|n〉ρnn〈n| , (1.12)

which is called the diagonal ensemble. It neglects all the off-diagonal terms, and the

diagonal terms are set by the initial condition. As being two special cases of the

diagonal ensemble, we also have the canonical ensemble as appears in the standard

equilibrium statistical mechanics

ρeq =1

Ze−βH =

1

Z

∑

n

|n〉e−βEn〈n| (1.13)

and the single many-body eigenstate “ensemble”

ρ(n) = |n〉〈n| (1.14)

as a limit of the standard microcanonical ensemble by sending the available energy

window ∆E → 0. When ETH is true, all different forms of the diagonal ensembles

7

including the two special cases above are equivalent for small subsystems, provided

the uncertainty in the total energy remains less than extensive.

1.3 Many-body Localization

As mentioned in the previous section, there exists a large class of systems, localized

systems, which do not obey ETH. The idea of localization first came from Anderson [1]

in 1958. In this section, I will focus on one subset of the localized systems: interacting

many-body localized (MBL) systems.

To briefly demonstrate the idea of many-body localization, I use the model of Pal

and Huse [20]. It is a one dimensional spin chain with spin-1/2 with Hamiltonian:

H =∑

i

hiσzi +

∑

i

Jσi · σi+1 , (1.15)

where σi are the Pauli operators for the spin-1/2 at ‘site’ i. The onsite magnetic

fields hi are static random variables with a probability distribution that is uniform in

[−h, h].

At J = 0, the many-body eigenstates are just the product states which are the

basis of the σz representation and the system is fully localized. For nonzero J , if

we assume J ≪ h, we can apply perturbation theory with zeroth order J = 0 to

construct the many-body eigenstates [5]. Because the local level spacing produced

by the J = 0 product states are comparable to h, it is in general much larger than

the interaction J . It implies that the eigenstates are very weakly hybridized. This

argument further implies that there is no DC spin transport or energy transport

and so the quantum thermalization is violated [5]. In addition to this perturbative

argument, Pal and Huse [20] demonstrated numerical evidence for the generic non-

perturbative case by doing the exact diagonalization up to 16 spins. They showed

8

that when h/J > hc/J ≈ 3.5, the system fails to behave thermal and goes into the

localized phase.

There are several properties that differ between thermalization and localization.

Some are summarized in Table 1.1 cited from [19]. Note that these differences are all

in dynamical properties or properties of the exact eigenstates. In fact, the two phases

do not differ at all in their static thermal equilibrium properties.

Thermal Phase Many-body Localized Phase

Memory of initial conditions ‘hid-den’ in global operators at longtimes

Some memory of local initial con-ditions preserved in local observ-ables at long times

ETH true ETH false

May have non-zero DC conduc-tivity

Zero DC conductivity

Continuous local spectrum Discrete local spectrum

Eigenstates with volume-law en-tanglement

Eigenstates with area-law entan-glement

Power-law in time spreading ofentanglement from non-entangledinitial condition

Logarithmic in time spreading ofentanglement from non-entangledinitial condition

Dephasing and dissipation Dephasing but no dissipation

Table 1.1: A list of some properties of the many-body localized phase contrasted withproperties of the thermal phases. Table from [19].

I close this section by mentioning the definition of temperature in the MBL phase.

As we know in the standard statistical mechanics, temperature is a crucial concept

in the equilibrium state. But in the MBL phase, the system fails to obey the ETH

hence the system cannot reach the equilibrium state. So temperature is ill-defined in

the MBL phase. One way to define temperature is to say that it is the corresponding

temperature if the system could thermalize. This implies a brand new type of phase

transition called the eigenstate phase transition and it will be discussed in the next

section.

9

1.4 Eigenstate Phase Transition

The eigenstate phase transition, as formulated e.g. in [13], is a brand new type

of phase transition compared with traditional thermal phase transition. In thermal

phase transition, we see a sharp change in terms of the thermal equilibrium when we

go from one phase to another. For example, from water to ice, the sharp change can

be observed directly by solving the minimal value of the free energy in the standard

statistical mechanical way and the minimal point of the free energy just corresponds to

the thermal equilibrium. However, in eigenstate phase transitions, the sharp change is

in properties of the many-body eigenstates but not the thermal equilibrium because

thermal equilibrium is an average over lots of eigenstates and this average washes

out the sharp change in terms of single many-body eigenstates. Thus, eigenstate

phase transition is invisible to the equilibrium statistical mechanics. For example,

as I mentioned in the previous section, the phase transition between the thermal

phase and the many-body localized phase is an eigenstate phase transition. In the

MBL phase, the system cannot even evolve to its thermal equilibrium value and some

information is still stored locally for a long time. In fact, if one would use the standard

equilibrium statistical mechanics to solve the MBL system, one could still get some

result and there would be no sharp change between the two phases in this sense.

The traditional equilibrium statistical mechanics however, does not correctly give the

long-time behavior in the MBL phase.

Because the eigenstate phase transition is a brand new type of phase transition,

it by itself has become a topic of research. Also, the usual equilibrium statistical

mechanics may break down when applied to the eigenstate phase transition, so the

properties of the critical point in the eigenstate phase transition, for example the

MBL transition, need to be reestablished and many questions about it are still open

[19]. In addition, eigenstate phase transition is not limited to the MBL transition.

Even in the regime where ETH is true on both sides, there remain eigenstate phase

10

transitions, at least in the limit of long-ranged interaction [32]. These are one of the

main topics in this thesis.

1.5 Thesis Outline

In this thesis, I will discuss two examples of eigenstate phase transitions as men-

tioned in the previous section. In Chapter 2, we explore a transverse-field Ising

model that exhibits both spontaneous symmetry-breaking and eigenstate thermaliza-

tion. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any

large but finite-sized system are all Schrödinger cat states: superpositions of states

with ‘up’ and ‘down’ spontaneous magnetization. This model exhibits two eigenstate

phase transitions within its ferromagnetic phase: In the lowest-temperature phase the

magnetization can macroscopically oscillate by quantum tunneling between up and

down. The relaxation of the magnetization is always overdamped in the remainder

of the ferromagnetic phase, which is further divided into phases where the system

thermally activates itself over the barrier between the up and down states, and where

it quantum tunnels.

In Chapter 3, we study the many-body localization transition in one dimensional

systems via the strong disorder renormalization group approach. In this framework,

we impose a set of rules for coarse-graining the system. The result from this set

of rules turns out to be a beautifully simple “toy” renormalization group. We can

almost solve for the critical fixed point distribution analytically. In addition, we also

get an estimate of the correlation length critical exponent ν ≈ 2.5 by both solv-

ing the analytical equations numerically and directly simulating the coarse-graining

procedure.

11

Chapter 2

Three ‘Species’ of Schrödinger Cat

States in an Infinite-range Spin

Model

2.1 Introduction

The dynamical properties of isolated many-body quantum systems have long been

of interest, due to their role in the fundamentals of quantum statistical mechanics.

More recently, experiments approximating this ideal of isolated many-body quantum

systems have become feasible in systems of trapped atoms [17, 8] and ions [7], and

as a consequence this topic has received renewed attention. It appears that a broad

class of such systems obey the Eigenstate Thermalization Hypothesis (ETH) [10, 26,

27, 22, 19]. The ETH asserts that each exact many-body eigenstate of a system’s

Hamiltonian is, all by itself, a proper microcanonical ensemble in the thermodynamic

limit, in which any small subsystem is thermally equilibrated, with the remainder of

the system acting as a reservoir. In the present chapter we present some interesting

12

results for an infinite-range transverse-field Ising model that obeys the ETH and also

has spontaneous symmetry-breaking.

Quantum many-body systems with static randomness may fail to obey the ETH

due to many-body Anderson localization stopping thermalization [1, 5, 20]. The

interesting interplay of many-body localization and discrete symmetry-breaking was

recently explored in Refs. [13, 21, 29, 18]; the present chapter instead explores an

example of the interplay of the ETH and Ising symmetry breaking.

We start with the infinite-range transverse-field Ising model:

H0 = −1

N

N∑

1=i

unchanged and can be used in our analysis. This randomness is too weak to produce

any localization. We have chosen to put the randomness on the interactions, since we

have found in exact diagonalizations that this produces much better thermalization

at numerically accessible system sizes as compared to, e.g., only making the local

transverse fields random.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7�0.35

�0.3

�0.25

�0.2

�0.15

�0.1

�0.05

0

Γ

u

Para

F3F1

Ground State

Thermodynamic PT

Eigenstate PT

QCP

F2

Figure 2.1: The phase diagram of our model. u is the energy per spin and Γ is thetransverse field. The ground state (zero temperature) is indicated by the black (solid)line, with the quantum critical point (QCP) indicated. The green (dot-dashed) line isthe thermodynamic phase transition (PT) between the paramagnetic and ferromag-netic (F) phases. There are two dynamical (eigenstate) phase transitions within theferromagnetic phases, indicated by the blue (dashed) lines. See the text for discussionsof the sharp distinctions between these three phases F1, F2 and F3.

We now briefly summarize our results, before deriving and discussing them in

more detail below. The phase diagram of this spin model as a function of the energy

14

per spin u = 〈H〉/N and the transverse field Γ is shown in Fig 2.1. There are the

usual two thermodynamic phases of a ferromagnet: the paramagnetic phase (Para)

at high energy and/or high |Γ|, and the ferromagnetic phase (F) when both |Γ| and

the energy are low enough. In the ferromagnetic phase, for any finite N , we can

ask about the dynamics of the system’s order parameter. There are three regimes

of behavior that are sharply distinguished from one another in the thermodynamic

limit N → ∞: At the highest energies within region F3 of the ferromagnetic phase

the system is a sufficiently large thermal reservoir for itself so that the most probable

path by which it flips from ‘up’ to ‘down’ magnetization under unitary time evolution

is by thermally activating itself over the free energy barrier between the two ordered

states. At lower energies (F1 and F2) the barrier is higher and wider and as a

result the reservoir is inadequate, so the system quantum tunnels through the barrier

when it flips the Ising order parameter. At the lowest energies in region F1 one

can in principle prepare a state that is a linear combination of two Schrödinger cat

eigenstates of H that will coherently oscillate via macroscopic quantum tunneling

between up and down magnetizations. In the intermediate energy regime (F2) the

magnetization dynamics due to quantum tunneling is always overdamped.

Throughout the ferromagnetic phase, the exact eigenstates of H for any finite N

are Schrödinger cat states that are superpositions of up and down magnetized states,

and the properties of these cats differ in the three regimes of the ferromagnetic phase

that are indicated in Fig 2.1. Thus the two phase transitions between these three

dynamically distinct ferromagnetic phases are not only dynamical phase transitions

but also ‘eigenstate phase transitions’ [13], while the equilibrium thermodynamic

properties are perfectly analytic through these two phase transitions.

We consider the infinite-range model not only because this allows a controlled

calculation of this novel physics within the ferromagnetic phases, but also because

finite-range, finite-dimensional models obeying ETH do not show these features. In

15

the latter models the free energy needed to flip the magnetization by making a domain

wall and sweeping it across the system is sub-extensive, while at any nonzero tem-

perature the system is a reservoir of extensive size, so a large system will always flip

via the thermal process without macroscopic quantum tunneling through the energy

barrier; phases F1 and F2 thus do not exist for such models. One can also consider

intermediate cases of transverse-field Ising models with interactions that fall off as a

power of the distance between spins. When this power is small enough, the resulting

free energy barrier to flip the magnetization in the ferromagnetic phase is extensive,

and we thus expect phases F1 and F2 to also occur in those models, although we do

not see a way to simply calculate the locations of the phase boundaries as we can for

the infinite-range model.

2.2 Infinite Range Transverse Field Ising Model

2.2.1 Integrability

First we examine the the unperturbed Hamiltonian H0, which has the same ther-

modynamics as our full model H. The Hamiltonian H0 commutes with all permuta-

tions of the spins, as does the total spin operator:

S ≡N∑

i=1

si . (2.3)

The magnitude S2 of the total spin squared also commutes withH0, so we can choose a

set of eigenstates of H0 that are also eigenstates of S2. This unperturbed Hamiltonian

16

only depends on the total spin and thus can be written as

H0 = −1

N

N∑

1=i

is the total spin value, then it is shown1 that

fN(S) = CN2−S

N − CN2−S−1

N . (2.5)

Then the entropy per spin Σ(S/N) defined by

Σ(S

N) ≡ 1

Nln fN(S) (2.6)

is given by2

Σ(S

N) = −

[

(1

2− S

N) ln(

1

2− S

N) + (

1

2+

S

N) ln(

1

2+

S

N)]

, (2.7)

due to all the different ways one can add together N spin-1/2’s to get total spin S

depends only on the ratio S/N .

For each value of S the spectrum of H0 has (2S+1) eigenenergies. These eigenen-

ergies and the corresponding eigenstates can be approximated for large S using a

discrete version of the WKB method [9], as we discuss below.

2.2.3 Discrete WKB Method

After solving the degeneracy of the energy level, the remaining task we need to

deal with is one copy of the block diagonal Hamiltonian with the total spin S. The

subspace is 2S + 1 dimensional and the Hamiltonian in the subspace is

H0 = −1

2NS2z − ΓSx

= − 12N

S2z −Γ

2(S+ + S−) , (2.8)

1Please see Appendix A for the mathematical proof.2Please see the end of Appendix A for the proof.

18

where S+, S− are raising and lowering operators. In (S2, Sz) representation, we have

S2z |S,m〉 = m2|S,m〉

S+|S,m〉 =√

(S −m)(S +m+ 1)|S,m+ 1〉

S−|S,m〉 =√

(S +m)(S −m+ 1)|S,m− 1〉 .

(2.9)

If we set the eigenstate wave function as

|ψ〉 =S∑

m=−SCm|S,m〉 , (2.10)

we arrive at the discrete version of the Schrödinger equation:

− 12N

m2Cm −Γ

2

√

(S +m)(S −m+ 1)Cm−1 −Γ

2

√

(S −m)(S +m+ 1)Cm+1 = E · Cm .

(2.11)

By applying the discrete WKB method reviewed by P.A.Braun[9], we define

wm =1

2Nm2

pm =Γ

2

√

(S +m)(S −m+ 1) ,(2.12)

then the Schrödinger equation reduces to the standard form in [9]:3

pmCm−1 + (wm + E)Cm + pm+1Cm+1 = 0 . (2.13)

By following the steps mentioned in [9], we introduce “momentum” ϕ = −i ∂∂m

, then

H0 = −(wm + pme−iϕ + pm+1eiϕ)

≃ −(wm + 2pm+ 12

cosϕ) . (2.14)

3In the original paper, the coefficient before Cm is “wm − E”, what we only need to do is toreplace all E’s into −E.

19

From now on, we will simplify the notation by denoting w ≡ wm and p ≡ pm+ 12

=

Γ

2

√

(S + 12)2 −m2. Then we introduce the “potential-energy curve”:4

U+(m) ≡ w + 2|p|

U−(m) ≡ w − 2|p| .(2.15)

The result given by [9] about U+ and U− is that the classical accessible energy region

is confined by U+ and U−:

−U+(m) ≤ E ≤ − U−(m) . (2.16)

In our model, the relation among U+, U− and E is demonstrated by Fig. 2.2. From

-U -

-U +

E

m_tm

U

Figure 2.2: Two “potential-energy curves” in our model.

Fig. 2.2, the classical accessible region is between the two curves, so if the energy is

specified as shown in Fig. 2.2, it would generically have the quantum tunneling effect

between the left and right well.

4In the original paper, the definition was U± ≡ w± 2p by assuming p > 0. Actually when p < 0,the situation is the same as p > 0 if we define U± ≡ w ± 2|p|.

20

Based on the connection relation associated by U+ turning point mt, we have the

WKB wave function[9]:

Cm>mt =A√vm

cos(

∫ m

mt

arccosB dm− π4

)

Cm

Then the turning point equation (2.20) reduces to:

1

2x2t + |Γ|

√

s̄2 − x2t + u = 0 ; (2.22)

⇒ xt =√2 ·

√

−u− Γ2 −√

Γ2(2u+ s̄2 + Γ2) . (2.23)

Meanwhile, the scaled logarithm of the tunneling rate γ reduces to:

γ =1

N

∫ mt

−mtarccosh

−E − 12N

m2

|Γ|√S2 −m2

dm

= 2

∫ xt

0

arccosh−u− 1

2x2

|Γ|√s̄2 − x2

dx . (2.24)

2.3 Thermodynamics

For the thermodynamics (but not the dynamics) of this system in the limit of large

N we can treat the components of S classically and ignore their nonzero commutators

when obtaining the extensive thermodynamic properties (energies, entropies, magne-

tizations). Then we obtain the energy density

u =E

N=

H0N

= −12s̄2z − Γs̄x , (2.25)

where

s̄x = ±√

s̄2 − s̄2y − s̄2z . (2.26)

Note that this equation is the same as the turning point equation (2.22) by setting

s̄y = 0 and s̄z = xt and choosing the sign of s̄x as the same as the sign of Γ. It

means that we can also use the semiclassical point of view to determine the WKB

22

turning point xt and further obtain the tunneling rate γ(s̄). The ground state is given

by minimizing the energy density u as a function of s̄, s̄y and s̄z. Here we choose

the sign of s̄x as the same as the sign of Γ so that the term −Γs̄x gives a negative

contribution to u. Since s̄ ≤ 1/2, it is easy to see that when s̄ = 1/2 and s̄y = 0,

u will be minimized in terms of them. By plugging these two conditions into u, we

have

u = −12s̄2z − |Γ|

√

1

4− s̄2z . (2.27)

By setting the first order derivative of s̄z equals zero, we have s̄z = 0 or s̄z =√

14− Γ2.

After evaluating the second order derivative, it becomes clear that when 14− Γ2 < 0,

s̄z = 0 is the minimal point; when14− Γ2 > 0, s̄z =

√

14− Γ2 minimizes u:

when |Γ| < 12, s̄z =

√

14− Γ2, umin = −18 − 12Γ2

when |Γ| > 12, s̄z = 0, umin = −12 |Γ|

. (2.28)

Thus, the ground state of H0 always has the maximum value of S = N/2. For

|Γ| ≥ 1/2, the ground state is paramagnetic with the spins polarized along the x-

direction: s̄y = s̄z = 0, s̄x =12sign(Γ) and u = 〈H0〉/N = −|Γ|/2. For |Γ| < 1/2, the

two nearly-degenerate ground states are ferromagnetic, with s̄z = ±(1/2)√1− 4Γ2,

s̄x = Γ, s̄y = 0 and u = −(1 + 4Γ2)/8. So the system has ground state quantum

phase transitions at Γ = ±1/2 between ferromagnetic phase (s̄z = 0, |Γ| < 1/2) and

paramagnetic phase (s̄z = 0, |Γ| > 1/2). The corresponding quantum critical points

are at Γc = ±1/2 , uc = −1/4.

For the excited states, the phase transition between ferromagnetic and paramag-

netic phase also exists. Since we are interested here in eigenstates, which are at a

given energy E = Nu, we will do the statistical mechanics in the microcanonical en-

semble. For a given transverse field Γ and energy E, the equilibrium (most probable)

23

state of the system is the one that maximizes the entropy, which means minimizing

the total spin S. To minimize S for a given E, clearly we set Sy = 0, since Sy does not

appear in the Hamiltonian. In the paramagnetic phase the total spin points along

the x-direction and the equilibrium value of the total spin is thus s̄eq = |u/Γ|. In

the ferromagnetic phase the system can go to higher entropy (lower total spin) for

a given u by making s̄z = 0. Some algebra shows that in the ferromagnetic phase,

which is |Γ| < 1/2 and −(1 + 4Γ2)/8 ≤ u < −Γ2, the equilibrium is at s̄x = Γ,

s̄z = ±√

2(−u− Γ2) and s̄eq =√−2u− Γ2. The line of critical points separating the

para- and ferromagnetic phases is u = −Γ2, for |Γ| < 1/2, as indicated in Fig 2.1; this

critical line ends at the quantum critical points at |Γ| = 1/2, u = −1/4. Note that

in this whole ferromagnetic phase regime, the sign of s̄x is the same as the sign of Γ.

Since all the quantities related to Γ, except s̄x, are even functions of Γ (or functions

of |Γ|), from now on we assume Γ > 0 and use Γ and |Γ| interchangeably. So it also

implies that s̄x ≥ 0.

2.4 Eigenstate Thermalization

Due to its full symmetry under all permutations of the N spins, the Hamiltonian

H0 is integrable, with all the good quantum numbers associated with this permutation

symmetry, including the magnitude S of the total spin. We want to study a more

generic system, so we add to the Hamiltonian the small term H1 (see Eq. (2.2))

to break the permutation symmetry, lift all degeneracies, and make the eigenstates

thermal. The only symmetry that remains in our full H is the Ising (Z2) symmetry

under a global rotation of all spins by angle π about their x axes.

For a given E and Γ, the eigenstates of H0 have total spin ranging from the

minimum and equilibrium value Seq up to the maximum value of S = N/2. To make

these in to thermal eigenstates we need H1 to perturb the system enough so that

24

the eigenstates of the full H are linear combinations of all these total spin values,

weighted as at thermal equilibrium. At first order, the perturbation we are adding,

H1, flips at most two spins, so it can change the total spin by at most ±2. The

spectrum of H0 at each value of total spin S contains (2S + 1) eigenenergies spread

over a range of energy that is of order S. Thus the level-spacing in the spectrum of

H0 at a given S remains of order one in the limit of large N . This is reflected in the

dynamics under H0, which is spin precession about the mean field, and the mean field

is of order one, so the rate of precession is also of order one.

For the eigenstates of H to strongly and thermally mix the different values of S

we thus need the matrix elements of H1 between states at different S to be large

compared to the (order-one) level spacing of H0 in the large N limit. This is why we

require that the exponent p in the definition of H1 satisfies p < 1, since this is the

condition for these matrix elements to diverge in the large N limit.5 This should be

sufficient to make all the eigenstates of H thermal in that limit. We have not yet

found a way to actually prove that this is sufficient to make all the eigenstates of H

satisfy the ETH, but below we provide some numerical evidence for this from exact

diagonalization of finite-size systems.

2.5 Dynamics

In addition to making sure that H1 is strong enough to thermalize the system, we

also want it to be weak enough so that we can use the well-understood dynamics and

thermodynamics of H0 in our analysis. By restricting the exponent p to be greater

than 1/2, in the large N limit the effective field that each spin is precessing about

is the mean field from H0, with only a small correction from H1 that vanishes as

N → ∞.6 This small correction is enough to thermalize the system for 1/2 < p < 1,5For the details of the order analysis of the lower bound of disorder, see Appendix B.1.6For the details of the order analysis of the upper bound of disorder, see Appendix B.2.

25

which is the range where the perturbation due to H1 on a single spin’s dynamics

vanishes for N → ∞, while the perturbation to the dynamics of the full many-body

system diverges. In this regime, the system’s primary dynamics is the S-conserving

dynamics due to H0, which is spin precession at a rate of order one, and, assuming

we are in the ferromagnetic phase, ‘attempts’ at rate of order one to tunnel through

the energy barrier between total Sz up and down. At a rate that is slower by a

power of N , the dynamics due to H1 allow ‘hopping’ between different values of the

total spin S, and thus thermalization to the equilibrium probability distribution of

S dictated by the entropy Σ. And at a rate that is slower still, exponentially slow

in N , the system succeeds in crossing the free energy barrier between up and down

magnetizations. It is the separation between these three time scales that allows us to

systematically understand the dynamics of this system for large N .

Since our full Hamiltonian H has Ising symmetry under a global spin flip, and

the randomness in H1 means there are no exact degeneracies in the spectrum of H,

for finite N any eigenstate of H is either even or odd under this Ising symmetry

(with probability one). In the ferromagnetic phase, this means the exact eigenstates

of H are all Schrödinger cat states that are either even or odd linear combinations

of states with total Sz up and down. These two equal (in magnitude) and opposite

(in sign) values of Sz are extensive, thus ‘macroscopically’ different, which is why it

is appropriate to call these eigenstates ‘Schrödinger cats’.

2.5.1 Thermal Activation and Quantum Tunneling

Next we examine the rate at which this system, in its ferromagnetic phase, will flip

from the up state with positive total Sz to the down state with negative Sz under the

unitary time evolution due to its Hamiltonian H. In Section 2.2.3, we noticed that

the system has an energy level splitting due to double-well quantum tunneling which

is given by ∆E ∼ e−γ(s̄)N where γ(s̄) is the scaled logarithm of the tunneling rate

26

as a function of the total spin density s̄ = S/N . By drawing the classically constant

energy line shown in Fig. 2.3 described by equation (2.25) by setting s̄y = 0 or (2.22)

E

s_0

x_t

s_min

s_z

s

Figure 2.3: Constant energy (E) line described by equation (2.25) by setting s̄y = 0or just by equation (2.22). It determines the WKB turning point xt when the totalspin density s̄ is fixed.

in the ferromagnetic phase region, we obtain a double-well shaped curve which has

a local maximum at s̄z = 0 where s̄ = s̄0 = −u/Γ and two local minima at |s̄z| = 0

where s̄ = s̄min = Seq/N .

There are two steps to this process: First the system gets ‘excited’ from its usual

(high entropy) total spin Seq ‘up’ to a larger total spin S with lower entropy, with a

probability ∼ exp {−N(Σ(Seq/N)− Σ(S/N))} given by the resulting decrease of the

entropy. As S is increased, the energy barrier, whose top is at energy E = −ΓS,

decreases. For E ≥ −ΓN/2, one way the system can flip is to increase S enough

so that E ≥ −ΓS and then it will simply cross over the top of the barrier without

quantum tunneling. In the higher-energy part (F3) of the ferromagnetic phase, in the

limit of large N this is the dominant process that flips the magnetization: the system

‘thermally activates’ itself (via its unitary time-evolution) to a low-entropy, high-

total-spin state where the energy barrier can be crossed without quantum tunneling.

27

The ‘height’ of the entropy barrier it must cross to do this is extensive:

N∆Σ = N(Σ(√−2u− Γ2)− Σ(−u/Γ)) , (2.29)

where ∆Σ is the reduction in entropy per site needed to go over the barrier.

If the system does not or can not go over the energy barrier by increasing the total

spin S, then in order to flip the magnetization it must quantum tunnel through the

barrier. For large N , this tunneling probability can be estimated using a version of the

WKB method [9] I have explained in Section 2.2.3. As a small summary of Section

2.2.3, the total Sz serves as the ‘position’, while the operator −ΓSx serves as the

‘kinetic energy’. What we need to calculate is the probability of the system tunneling

between positive and negative total Sz for a given total spin S = Ns̄ satisfying

Seq ≤ S < −E/Γ. This probability behaves as ∼ exp (−Nγ), with γ(u, s̄,Γ) being

an intensive quantity. If we define a scaled ‘position’ x = Sz/N , the WKB ‘turning

points’ adjacent to the barrier are at x = ±xt with

xt =√2 ·

√

−u− Γ2 −√

Γ2(2u+ s̄2 + Γ2) . (2.30)

Then the intensive factor in the exponent of the tunneling probability is given by the

WKB tunneling integral

γ = 2

∫ xt

0

arcosh−u− 1

2x2

Γ√s̄2 − x2

dx . (2.31)

Since the probabilities of being ‘excited’ to total spin S and of quantum tunneling

through the barrier with total spin S are both exponentially small in N , in the limit of

large N the dominant process by which the magnetization flips is given by a standard

‘saddle point’ condition. The total spin S = Ns̄ at which the system tunnels is the

value that maximizes the product of these two probabilities and thus minimizes the

28

quantity

α(u,Γ) = mins̄{Σ(s̄min =√−2u− Γ2)− Σ(s̄) + γ(u, s̄,Γ)} . (2.32)

This means in order to tunnel through the barrier, the system will finally choose one

particular value of the total spin s̄ to maximize {Σ(s̄)− Σ(s̄min)− γ(u, s̄,Γ)}. Then

the distinction between thermal and quantum tunneling becomes clear: if the total

spin is chosen to be in the interval (s̄min, s̄0) which is below the top of the barrier, it is

quantum tunneling; if the total spin is chosen to be bigger than or equal to s̄0 = −u/Γ

which is at the top of the barrier, it is thermal activation over the barrier.

The sketch of Σ(s̄) and γ(s̄) is shown in Fig. 2.4, where dγ(s̄)ds̄

diverges7 at s̄ = s̄min;

�(s)

Γ(s)

0 s_min s_0 1/2s

�, Γ

Figure 2.4: A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄).

γ(s̄) becomes zero at s̄ = s̄0 since it is at the top of the barrier. From Fig. 2.4, if

we assume γ(s̄) is a convex8 function of s̄, then the distinction between the thermal

activation and quantum tunneling would be shown straightforward in Fig. 2.5 where

in thermal activation, Σ − γ reaches its maximum at s̄ = s̄0; in quantum tunneling,

Σ− γ reaches its maximum in the middle between s̄min and s̄0. Then, the condition7Please see Appendix C for proof of the divergence.8For detailed discussion of this issue, see Appendix D.

29

s_0s_mins

��Γ

(a) Thermal activation

s_min s_0s

��Γ

(b) Quantum tunneling

Figure 2.5: A sketch of the difference between thermal activation and quantum tun-neling.

reduces to:

if ∂∂s̄(Σ− γ)

∣

∣

∣

s̄=s̄0−0> 0 ⇒ Thermal Activation;

if ∂∂s̄(Σ− γ)

∣

∣

∣

s̄=s̄0−0< 0 ⇒ Quantum Tunneling.

(2.33)

We have located the saddle point numerically at many points within the ferro-

magnetic phase and it appears to always be unique, without any discontinuities as

the parameters u and Γ are varied. Some straightforward analysis as I will discuss in

Section 2.6 in details shows that in the higher-energy part (F3) of the ferromagnetic

phase where

πΓ√−u− Γ2

≥ ln Γ− 2uΓ+ 2u

(2.34)

the saddle point is ‘thermal’: the ‘entropy cost’ of going to higher S is less than the

‘tunneling cost’, and the system goes over the barrier without any quantum tunneling.

We call the eigenstates in this regime ‘thermal cats’, since these Schrödinger cat states

flip by thermally activating themselves over the barrier. In regions F1 and F2 this

inequality is instead false, and the system quantum tunnels through the barrier at a

value of S satisfying Seq ≤ S < −E/Γ, so the eigenstates are instead ‘quantum cats’.

30

The location of the dynamical phase transition between phases F2 and F3 is given

by converting the above inequality (2.88) to an equality.

The asymptotic behavior of this transition curve will be discussed in details in

Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ =

1/2 − |Γ| ≪ 1), this transition line becomes exponentially adjacent to (and above)

the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) + O(exp(−1/√∆Γ)); near Γ = 0,

it behaves as a power law: −u ≈ (|Γ|√

π/4)4/3.

2.5.2 Paired States and Unpaired States

Within the lower-energy phases (F1 and F2) of ‘quantum cats’, there is a second

dynamical phase transition within the ferromagnetic phase. This is also a ‘spectral

phase transition’ [13] in the level-spacing statistics of the eigenenergies. If one starts

with a state (not an eigenstate) that is magnetized up, the rate at which the system

crosses the barrier to down is ∼ exp (−Nα(u,Γ)), and as a result the uncertainty of

the energy of this initial ‘up’ state must be at least this large, by the time-energy

uncertainty relation. Compare this minimum energy uncertainty to the typical many-

body level spacing ∼ exp (−NΣ(√−2u− Γ2)) of the eigenstates of H at that energy.

There is clearly a sharp change at the phase transition line, which is the line where

α(u,Γ) = Σ(√−2u− Γ2). In another word,

if maxs̄

(Σ− γ) < 0 ⇒ Paired States;

if maxs̄

(Σ− γ) > 0 ⇒ Unpaired States.(2.35)

This transition line between phases F1 and F2 is shown in Fig 2.1. The location of

this transition was obtained numerically, since we do not have a simple closed-form

expression for α(u,Γ).

The asymptotic behavior of this transition curve will also be discussed in details

in Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ =

31

1/2− |Γ| ≪ 1), this transition line also becomes exponentially adjacent to (but now

below) the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) − O(exp(−1/√∆Γ)); near

Γ = 0, it is logarithmically tangent to the u-axis: −u ∼ 1/(ln |Γ|)2.

2.6 Analytical Calculation of the Thermal Activa-

tion and Quantum Tunneling Transition

The transition between thermal activation and quantum tunneling is illustrated

by condition (2.33). Thus, in order to find the transition, we need to calculate the

derivatives ∂Σ∂s̄

and ∂γ∂s̄

at s̄ = s̄0 − 0. The first one is straightforward from equation

(2.7):

∂Σ

∂s̄= − ln 1 + 2s̄

1− 2s̄ . (2.36)

Therefore we need to focus on solving ∂γ∂s̄

at s̄ = s̄0 − 0.

When s̄ = s̄0, by definition we have the turning point xt = 0 (It implies γ(s̄0) = 0.).

Plugging xt = 0 into the turning point equation (2.22) gives us:

s̄0 =−uΓ

. (2.37)

Now we perturb xt a little away from 0 by setting xt = ε ≪ 1. Then we have:

1

2ε2 + Γ

√s̄2 − ε2 + u = 0

Γ√s̄2 − ε2 = −u− 1

2ε2

s̄2 − ε2 = 1Γ2

(−u− 12ε2)2

s̄2 =u2

Γ2+ ε2 +

uε2

Γ2+O(ε4) .

32

Then,

⇒ s̄ =√

u2

Γ2+ ε2 +

uε2

Γ2+O(ε4)

=−uΓ

√

1 +u+ Γ2

u2ε2 +O(ε4)

=−uΓ

[

1 +u+ Γ2

2u2ε2 +O(ε4)

]

= −uΓ− u+ Γ

2

2uΓε2 +O(ε4) .

Thus the infinitesimal change of s̄ is:

∆s̄ = −u+ Γ2

2uΓε2 +O(ε4) < 0 . (2.38)

Now it is time to see ∆γ:

∆γ = γ(xt = ε) = 2

∫ ε

0

arccosh−u− 1

2x2

Γ√s̄2 − x2

dx . (2.39)

Since in this integral x runs from 0 to ε, we may do the rescaling to let x ≡ εb where

b runs from 0 to 1. After that, we have:

B =−u− 1

2ε2b2

Γ√s̄2 − ε2b2

=−u− 1

2ε2b2

Γ

√

u2

Γ2+ (1 + u

Γ2)ε2 − ε2b2 +O(ε4)

=−u− 1

2ε2b2

√

u2 + (u+ Γ2 − b2Γ2)ε2 +O(ε4)

=−u− 1

2ε2b2

(−u)√

1 + u+Γ2−b2Γ2u2

ε2 +O(ε4)

=1 + b

2

2uε2

1 + u+Γ2−b2Γ22u2

ε2 +O(ε4)

= 1 +( b2

2u− u+ Γ

2 − b2Γ22u2

)

ε2 +O(ε4) . (2.40)

33

Since arccoshB = ln(B +√B2 − 1), then if B = 1 +∆ where ∆ ≪ 1, we have

arccosh(1 +∆) = ln(1 +∆+√2∆+∆2)

= ln[1 +∆+√2∆+O(∆

√∆)]

= ∆+√2∆+O(∆

√∆)− 1

2[∆+

√2∆+O(∆

√∆)]2 +O(∆

√∆)

= ∆+√2∆− 1

2· 2∆+O(∆

√∆)

=√2∆+O(∆

√∆) . (2.41)

So if ∆ =(

b2

2u− u+Γ2−b2Γ2

2u2

)

ε2 +O(ε4), then

arccosh(1 +∆) =

√

b2

u− u+ Γ

2 − b2Γ2u2

ε+O(ε3) . (2.42)

⇒ γ(xt = ε) = 2∫ 1

0

√

b2

u− u+ Γ

2 − b2Γ2u2

ε · ε db+O(ε4)

= 2ε2∫ 1

0

1

−u√

ub2 − (Γ2 + u) + b2Γ2 db+O(ε4)

= 2ε2∫ 1

0

√−u− Γ2−u

√1− b2 db+O(ε4) . (2.43)

If we change the variable b = sin θ, then∫ 1

0

√1− b2 db =

∫ π

2

0cos θ ·cos θ dθ = π

4. Thus

we arrive at:

γ(xt = ε) = 2 ·

√−u− Γ2−u ·

π

4· ε2 +O(ε4) , (2.44)

34

∂γ

∂s̄

∣

∣

∣

s̄=s̄0−0= lim

ε→0

γ(xt = ε)− 0∆s̄

=2 ·

√−u−Γ2−u ·

π4

−u+Γ22uΓ

= − πΓ√−u− Γ2

. (2.45)

In addition,

∂Σ

∂s̄

∣

∣

∣

s̄=s̄0= − ln 1 + 2s̄0

1− 2s̄0= − ln Γ− 2u

Γ+ 2u. (2.46)

Therefore the boundary between thermal activation and quantum tunneling is given

by:

∂γ

∂s̄

∣

∣

∣

s̄=s̄0−0=

∂Σ

∂s̄

∣

∣

∣

s̄=s̄0

⇒ πΓ√−u− Γ2

= lnΓ− 2uΓ+ 2u

. (2.47)

To be more specific, we arrive at the result:

if πΓ√−u−Γ2 > lnΓ−2uΓ+2u

⇒ ∂∂s̄(Σ− γ)

∣

∣

∣

s̄=s̄0−0> 0 ⇒ Thermal Activation;

if πΓ√−u−Γ2 < lnΓ−2uΓ+2u

⇒ ∂∂s̄(Σ− γ)

∣

∣

∣

s̄=s̄0−0< 0 ⇒ Quantum Tunneling.

(2.48)

2.7 Asymptotic Behaviors of Both Transitions

Near Both QCP and u = 0 ,Γ = 0 Point

Up to now, we have already obtained the mathematical condition of both tran-

sition lines. To further explore the properties of both lines, it is better to also have

the asymptotic behaviors near the edges of the ferromagnetic regime demonstrated

35

by Fig. 2.1. One edge is the Quantum Critical Point(QCP); the other is the origin

of the phase diagram: u = 0 ,Γ = 0 point. We shall study the asymptotic behaviors

one by one.

2.7.1 Near QCP

The QCP is at Γc = 1/2 , uc = −1/4. We perturb the system a little away from

the QCP into the ferromagnetic region to let Γ = 12−∆Γ where ∆Γ ≪ 1. Then the

goal is to see what asymptotic behavior the energy density u should have in terms of

∆Γ. Let us discuss these two transitions one by one.

Thermal Activation and Quantum Tunneling Transition

In the thermal activation and quantum tunneling transition case, the transition

line must be placed above the straight line u = −Γ/2. If not, the top of the barrier

s̄0 =−uΓ

in Fig. 2.3 would be greater than 1/2 which is the upper bound of the

available total spin density s̄. Then there would be always a quantum tunneling

effect and no thermal activation can be achieved. So the transition line must be

placed in the regime where s̄0 =−uΓ

≤ 12or u ≥ −Γ/2. Also we know that u ≤ −Γ2

since it is in ferromagnetic region. Note that u = −Γ2 has a first order derivative −1

at Γc = 1/2. Therefore we set9

u = −14+ (

1

2+ δ) ·∆Γ , (2.49)

where 0 ≤ δ ≤ 1/2. Plugging this definition into the transition line equation:

πΓ√−u− Γ2

= lnΓ− 2uΓ+ 2u

, (2.50)

9This is only a redefinition of a variable but not a Taylor expansion since the essential singularityoccurs in general. And the same concern applies to all cases afterwards.

36

we then arrive at:

L.H.S. =π(1

2−∆Γ)

√

(12− δ)∆Γ−∆Γ2

� O( 1√

∆Γ

)

;

R.H.S. = ln1− 2(1 + δ)∆Γ

2δ∆Γ∼ O

(

− ln(δ∆Γ))

. (2.51)

Thus, if δ ∼ O(1), then the right hand side would be O(

− ln(∆Γ))

which is much

smaller than O(

1√∆Γ

)

. So in order to let the right hand side big enough, we need

δ ≪ 1. Under this condition, the left hand side is exactly O(

1√∆Γ

)

. By setting the

left hand side to be equal to the right hand side, we finally get:

1√∆Γ

∼ − ln(δ∆Γ)

⇒ δ ∼ 1∆Γ

e− 1√

∆Γ . (2.52)

Note that the form e− 1√

∆Γ behaves essentially singular at ∆Γ = 0 point and does

not have a proper Taylor expansion in terms of ∆Γ; in fact, it is much smaller than

any finite power of ∆Γ, so all the Taylor coefficients would be zero. It implies the

transition line is extremely near (and above) the straight line u = −Γ/2. So finally,

the asymptotic behavior of the thermal activation and quantum tunneling transition

is:

Γ =1

2−∆Γ

u = −14+

1

2∆Γ+O

(

e− 1√

∆Γ

)

,

(2.53)

where ∆Γ ≪ 1.

37

Paired and Unpaired States Transition

In the paired and unpaired states transition case, on the contrary, the transition

line must be placed below the straight line u = −Γ/2. If not, then s̄0 = −uΓ would be

an available candidate for taking the maximal value of Σ(s̄) − γ(s̄). By noting that

γ(s̄0) = 0, we would have

maxs̄

(Σ− γ) ≥ Σ(s̄0)− γ(s̄0)

= Σ(s̄0) > 0 , (2.54)

then it would be always in unpaired states region. So the transition line must be

placed in the regime where u ≤ − Γ/2. Also we know that u ≥ − 18− 1

2Γ2 since the

right hand side is the ground state energy. Note that at Γc = 1/2, u = −18 − 12Γ2 has

a first order derivative −1/2 which is the same as that of the straight line u = −Γ/2.

Since the transition line must be placed between these two bounds, we obtain the

first order derivative of the transition line must also be −1/2. Therefore we set

u = −14+

1

2∆Γ− λ1∆Γ2 , (2.55)

where 0 ≤ λ1 ≤ 1/2.

In order to solve maxs̄

(Σ − γ), we need to first solve the upper and lower bound

of the available s̄. The upper bound is 1/2 since in this case s̄0 ≥ 1/2. The lower

bound is exactly s̄min indicated in Fig. 2.3. By solving s̄ in the turning point equation

(2.22), we arrive at:

s̄2 =u2

Γ2+ (1 +

u

Γ2)x2t +

1

4Γ2x4t , (2.56)

38

then the minimal available s̄2 is exactly

s̄2min = −2u− Γ2 (2.57)

at (xt)2min = −2u − 2Γ2. By plugging the definition of u and Γ in terms of ∆Γ and

λ1 into s̄2min, we have:

s̄2min = −2u− Γ2 =1

4− (1− 2λ1)∆Γ2 . (2.58)

Then we define

s̄2 ≡ 14− (1− 2λ1) · α1 ·∆Γ2 , (2.59)

where 0 ≤ α1 ≤ 1. Then the task converts into finding the maximal value of Σ − γ

in terms of α1.

Since ∆Γ ≪ 1 and λ1 ,α1 � O(1), we have

s̄ =1

2− (1− 2λ1)α1∆Γ2 +O

(

[

(1− 2λ1)α1∆Γ2]2)

(2.60)

⇒ Σ(s̄) = −[

(1

2− s̄) ln(1

2− s̄) + (1

2− s̄) ln(1

2− s̄)

]

= −[

(1− 2λ1)α1∆Γ2 ln(

(1− 2λ1)α1∆Γ2)

+O(

(1− 2λ1)α1∆Γ2)]

� −∆Γ2 ln(∆Γ2) . (2.61)

Then for

γ(s̄) = 2

∫ xt

0

arccosh−u− 1

2x2

Γ√s̄2 − x2

dx

= 2

∫ 1

0

(

arccosh−u− 1

2x2t b

2

Γ√

s̄2 − x2t b2)

xt db , (2.62)

39

we have:

x2t = 2 ·(

− u− Γ2 − Γ√2u+ s̄2 + Γ2

)

= 2 ·[1

4− 1

2∆Γ+ λ1∆Γ

2 − (12−∆Γ)2

− (12−∆Γ)

√

1

4− (1− 2λ1)α1∆Γ2 −

1

4+ (1− 2λ1)∆Γ2

]

= 2 ·[(1

2− 1

2

√

(1− 2λ1)(1− α1))

∆Γ+(

λ1 − 1 +√

(1− 2λ1)(1− α1))

∆Γ2]

≡ 2 · (a1∆Γ+ a2∆Γ2) , (2.63)

where a1 ≡ 12 − 12√

(1− 2λ1)(1− α1) , a2 ≡ λ1 − 1 +√

(1− 2λ1)(1− α1) . Then, we

have:

−u− 12x2t b

2

Γ√

s̄2 − x2t b2

=14− 1

2∆Γ+ λ1∆Γ

2 − (a1∆Γ+ a2∆Γ2)b2

(12−∆Γ)

√

14− (1− 2λ1)α1∆Γ2 − 2(a1∆Γ+ a2∆Γ2)b2

≈[1

2+

λ1∆Γ2 − a1b2∆Γ− a2b2∆Γ2

12−∆Γ

]

· 2[

1 + 2(1− 2λ1)α1∆Γ2 + 4a1b2∆Γ2 + 4a2b2∆Γ2]

= 1 +2(λ1∆Γ

2 − a1b2∆Γ− a2b2∆Γ2)12−∆Γ + 2(1− 2λ1)α1∆Γ

2 + 4a1b2∆Γ+ 4a2b

2∆Γ

2

= 1 +1

12−∆Γ

[

2λ1∆Γ2 − 2a1b2∆Γ− 2a2b2∆Γ2 + (1− 2∆Γ)(1− 2λ1)α1∆Γ2

+ 2a1b2∆Γ− 4a1b2∆Γ2 + 2a2b2∆Γ2 − 4a2b2∆Γ3

]

≈ 1 + 2∆Γ2[

2λ1 + (1− 2λ1)α1 − 4a1b2 − 4a2b2∆Γ]

, (2.64)

where all the terms I have omitted are definitely much smaller than at least one of

the present terms in the last line given ∆Γ ≪ 1 ,λ1 � O(1) ,α1 � O(1) .

Next comes the key point: in order to have Σ and γ in same order at some

value of α1, both α1 and λ1 have to be much smaller than 1. Let me prove it by

contradiction. If the conclusion is not true, then at least one of α1 or λ1 would be

40

O(1). Then, we would have both a1 ∼ O(1) and a2 ∼ O(1). So xt ∼√∆Γ and

−u− 12x2t b

2

Γ

√s̄2−x2t b2

∼ 1 + O(∆Γ2). Since we know arccosh(1 + ∆) =√2∆ + O(∆

√∆) when

∆ ≪ 1, it means that:

γ(s̄) = 2

∫ 1

0

(

arccosh−u− 1

2x2t b

2

Γ√

s̄2 − x2t b2)

xt db

∼√∆Γ2 ·

√∆Γ

= ∆Γ3

2 . (2.65)

However, from equation (2.61), we have Σ � ∆Γ2 ln(∆Γ2). So we would arrive at

Σ(s̄) ≪ γ(s̄) which is not the condition of the transition line. Thus, by contradiction,

we conclude both α1 ≪ 1 and λ1 ≪ 1 have to be true in order to satisfy the transition

condition.

By knowing α1 ≪ 1 and λ1 ≪ 1, all quantities are further simplified. First, we

have:

a1 ≈1

2λ1 +

1

4α1 , a2 = λ1 − 2a1 ≈ −

1

2α1 , (2.66)

which are both linear in λ1 and α1. Then, we get:

xt ≈√

2a1∆Γ ≈√

(λ1 +1

2α1)∆Γ , (2.67)

and

−u− 12x2t b

2

Γ√

s̄2 − x2t b2≈ 1 + 2(2λ1 + α1 − 4a1b2)∆Γ2

≈ 1 + 2(2λ1 + α1)(1− b2)∆Γ2 . (2.68)

41

⇒ γ(s̄) ≈ 2∫ 1

0

√

2 · 2(2λ1 + α1)(1− b2)∆Γ2√

(λ1 +1

2α1)∆Γ db

= 2√2(2λ1 + α1)∆Γ

3/2

∫ 1

0

√1− b2 db

=

√2

2π(2λ1 + α1)∆Γ

3/2 . (2.69)

In addition, from equation (2.61) we have:

Σ(s̄) ≈ − (1− 2λ1)α1∆Γ2 ln(

(1− 2λ1)α1∆Γ2)

≈ α1∆Γ2 ln(

α1∆Γ2)

. (2.70)

In order to let Σ and γ be in same order, we finally arrive at:

(2λ1 + α1)∆Γ3/2 ∼ α1∆Γ2 ln

(

α1∆Γ2)

. (2.71)

Then it is better to let O(λ1) ≤ O(α1), since if not, both λ1 and α1 would have an

extra factor λ1/α1 on the exponent which would make the value far smaller than the

case O(λ1) ≤ O(α1). Therefore we finally get the result:

α1 ∼1

∆Γ2e− 1√

∆Γ , λ1 ∼1

∆Γ2e− 1√

∆Γ . (2.72)

They also behave essentially singular at ∆Γ = 0 point so the transition line is also

extremely near (but this time below) the straight line u = −Γ/2. Finally, the asymp-

totic behavior of the paired and unpaired transition is also:

Γ =1

2−∆Γ

u = −14+

1

2∆Γ+O

(

e− 1√

∆Γ

)

,

(2.73)

where ∆Γ ≪ 1.

42

2.7.2 Near u = 0 ,Γ = 0 Point

The other interested point is the origin of the phase diagram: u = 0 ,Γ = 0 point.

Also, we will perturb the system a little away from Γ = 0 or equivalently let Γ ≪ 1.

Then the goal is to see what u behaves in terms of Γ asymptotically when Γ → 0.

Thermal Activation and Quantum Tunneling Transition

In this case, we know that Γ2 ≤ − u ≤ Γ/2. First consider two extreme cases:

1. If −u ∼ O(Γ), then πΓ√−u−Γ2 ∼ O(√Γ) , ln

(

Γ−2uΓ+2u

)

� O(1). So we have

πΓ√−u−Γ2 ≪ ln

(

Γ−2uΓ+2u

)

. It is in thermal activation region.

2. If −u ∼ O(Γ2), then πΓ√−u−Γ2 ∼ O(1) , ln(

Γ−2uΓ+2u

)

∼ O(Γ). Then we haveπΓ√

−u−Γ2 ≫ ln(

Γ−2uΓ+2u

)

. It is in quantum tunneling region.

Thus, the transition line must be placed in between where Γ2 ≪ − u ≪ Γ. When

this is true,

πΓ√−u− Γ2

≈ πΓ√−u ,

ln(

Γ− 2uΓ+ 2u

)

≈ −4uΓ

. (2.74)

⇒ πΓ√−u =−4uΓ

,

π

4Γ2 = (−u) 32 ,

−u =(

√

π

4Γ

) 4

3 ∼ Γ 43 . (2.75)

This is a power law behavior when Γ → 0.

Paired and Unpaired States Transition

In this case, we have Γ/2 ≤ −u ≤ 1/8. Also consider the two extreme cases first:43

1. If −u ∼ O(1), then xt =√2√

−u− Γ2 − Γ√2u+ s̄2 + Γ2 ≈

√−2u ∼ O(1).

But B =−u− 1

2x2t b

2

Γ

√s̄2−x2t b2

∼ O(−uΓ) → ∞. In addition, arccoshB = ln(B +

√B2 − 1) → ln 2B when B → ∞. So γ = 2

∫ 1

0

(

arccosh−u− 1

2x2t b

2

Γ√

s̄2 − x2t b2)

xt db ∼

− lnΓ → ∞. But Σ � O(1), so it means maxs̄

(Σ− γ) < 0. It is in paired states

region. Thus the transition line tends to reach u = 0 point when Γ → 0.

2. If −u ∼ O(Γ), then we define −u = α2Γ, where α2 ∼ O(1) and α2 ≥ 1/2 since

we have −u ≥ Γ/2. Under this assumption, we have:

xt =√2

√

−u− Γ2 − Γ√2u+ s̄2 + Γ2

=√2

√

α2Γ− Γ2 − Γ√

s̄2 − 2α2Γ+ Γ2

≈√2

√

α2Γ− Γ√

s̄2 − 2α2Γ

=√2 ·

√Γ ·

√

α2 −√

s̄2 − 2α2Γ

� O(√Γ) , (2.76)

whereas,

B =−u− 1

2x2t b

2

Γ√

s̄2 − x2t b2

≈ α2Γ− b2Γ(α2 −

√s̄2 − 2α2Γ)

Γ√

s̄2 − x2t b2

=α2 − b2(α2 −

√s̄2 − 2α2Γ)

√

s̄2 − x2t b2. (2.77)

So for a given finite s̄, we have B ∼ O(1). Thus, γ = 2∫ 1

0

(arccoshB)xt db �

O(√Γ) → 0, but Σ ∼ O(1), so it means max

s̄(Σ − γ) > 0. It is in unpaired

states region.

Thus, the transition line must also be placed in between where Γ ≪ − u ≪ 1.

Then we get xt =√2√

−u− Γ2 − Γ√2u+ s̄2 + Γ2 ∼ O(

√−u) and B = −u−

1

2x2t b

2

Γ

√s̄2−x2t b2

∼44

O(−uΓ) ≫ 1.

⇒ γ = 2∫ 1

0

(arccoshB)xt db ∼ O[

ln(−u

Γ

)

·√−u

]

. (2.78)

In order to let γ comparable to Σ which is typically O(1), we should have:

O[

ln(−u

Γ

)

·√−u

]

∼ O(1)

⇒(

ln(−u)− lnΓ)√

−u ∼ 1

⇒ − lnΓ√−u ∼ 1

⇒ − u ∼ 1(lnΓ)2

. (2.79)

This is equivalent to Γ ∼ e−1√−u which is also an essential singularity at u = 0 ,Γ = 0

point. It means that the transition line is extremely near the u-axis but not the

u = −Γ/2 line.

2.8 Numerical Evidence

We now present some numerical results about the thermalization properties and

the distinction between phases F1 and F3 (phase F2 is too narrow to clearly see in

the size systems we can diagonalize). We exactly diagonalize Hamiltonians with up

to N = 15 spins. We first put the Hamiltonian into a block diagonal form using basis

states that are even and odd with respect to the system’s Z2 Ising symmetry, and then

diagonalize each sector numerically to obtain all of the eigenstates within that sector.

For finite N , the coefficient λ in the disorder term H1 in Eq. (2.2) matters. If λ is too

large the system will become a spin glass rather than a ferromagnet. Thus λ needs to

be carefully chosen to be large enough for our finite systems to show thermalization,

45

but small enough to avoid the spin glass regime. After some exploration, we chose to

use the parameters λ = 0.7, p = 3/4 and Γ = 1/8 for our exact diagonalizations.

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

δ

f

Figure 2.6: The level-spacing statistics using 100 realizations of H at N = 15 in phaseF1 within the even sector. δ is the ratio between the smaller level spacing δ< tothe larger level spacing δ> for three consecutive eigenenergies in the even sector. f isthe relative frequency for each bin in this histogram.

First, we show the level-spacing statistics, which should be GOE if the eigenstates

are thermal. This must be done within one Z2 symmetry sector, since there is no level-

repulsion between states in different sectors. We look at each set of three consecutive

levels in one sector and denote δ< as the smaller level spacing and δ> as the larger

level spacing. Then the histogram of the ratio δ can be compared to GOE level

statistics [2]. The even sector results in the F1 phase for 100 realizations at N = 15

are shown in Fig. 2.6. We see the expected strong level repulsion, consistent with the

46

thermalization. All other phases and symmetry sectors were also examined and the

results are also thermal, since phase F1 is the lowest-energy phase and thus the most

difficult to thermalize.

Next, we examine a ‘distance’ between two eigenstates that are adjacent in the

energy spectrum by comparing their probability distributions for the total spin Sz.

We define this distance between eigenstate 1 and eigenstate 2 as

D12 =

N/2∑

Sz=−N/2

∣

∣P1(Sz)− P2(Sz)∣

∣ , (2.80)

where P1(Sz) and P2(Sz) are the probability distributions of Sz in eigenstates 1 and

2, respectively. We tested three different distances: Deo the distance between an even

parity state (eigenstate 1) and the nearest-energy odd parity state (eigenstate 2), and

similarly for Dee and Doo. If a system thermalizes, each eigenstate is equivalent to

a microcanonical ensemble characterized by its energy. For two eigenstates that are

adjacent in energy, the energy difference ∆E ∼ 2−N , therefore we expect the eigen-

state distances Deo, Dee and Doo should decrease exponentially with N . In phase F1,

since the spectrum consists of nearly-degenerate pairs of states, the energy differences

satisfy ∆Eeo ≪ ∆Eee , ∆Eoo. Thus we expect in phase F1, Deo ≪ Dee , Doo. In addi-

tion, if we choose the upper bound of the energy window we average over to be well

within the F1 phase, we would also expect that Eq. (2.84) holds, so the exponential

decay rate of Deo would be greater than those of Dee and Doo. Meanwhile, in phases

other than F1, we expect all three D’s are well coincident. The numerical results are

shown in Fig. 2.7. As we expected, all these eigenstate distances decay exponentially

with N . In addition, Deo in phase F1 is much smaller and decreasing much faster

than the other two distances, and in phase F3 all three D’s are well coincident. This

demonstrates the clear distinction between phases F1 and F3, and further tests the

thermalization in phase F3.

47

8 9 10 11 12 13 14 15

�6

�5

�4

�3

�2

�1

N

log(D

)

e�o

e�e

o�o

(a) Eigenstate distance in phase F1

8 9 10 11 12 13 14 15

�3.5

�3

�2.5

�2

�1.5

N

log(D

)

e�o

e�e

o�o

(b) Eigenstate distance in phase F3

Figure 2.7: Averages of log (D) in phases F1 and F3, respectively, where D is the‘eigenstate distance’ defined in the text.. The energy density range we used in F1is from the first excited state in each sector up to uc − 0.02 where uc is the energydensity at the phase boundary between F1 and F2, whereas in F3 we used the phase’sfull energy density range. N is the total number of spins varying from 8 to 15. Theexponential decrease of D with increasing N indicates thermalization. The error barscome from averaging over 100 realizations.

48

�0.14 �0.12 �0.1 �0.08 �0.06 �0.04 �0.02 00

0.2

0.4

0.6

0.8

1

1.2

F1

F2

F3

u

αn

ᾱn at N = 11

ᾱn at N = 15

Δαn at N = 11Δαn at N = 15

N → ∞

Figure 2.8: The mean ᾱn and the standard deviation ∆αn of the quantity αn definedin Eq. (2.81). The number of realizations is 1600 for N = 11 (blue dash-dotted lines)and 100 for N = 15 (red dashed lines). The green (solid) line gives the theoreticalquantity α(u,Γ) defined in Eq. (2.32) for the system size N → ∞.

In the ferromagnetic phases, the system spontaneously flips between magnetiza-

tion up and down at a rate that behaves as ∼ exp(−Nα(u,Γ)), where the quantity

α(u,Γ) is defined in Eq. (2.32). The probability of the system having total magneti-

zation zero (or 1/2 for systems with odd N) also behaves as ∼ exp(−Nα(u,Γ)). Thus

from a single many-body eigenstate |n〉 we can obtain an estimate of the quantity α

as

αn =1

Nln

maxSz{Pn(S